The approach that could be followed so as to attempt making the chromatic aberration of an imaging system equal to zero by means of optical elements which are not rotationally symmetrical has already been described in an article entitled "Spharische und chromatische Korrektur von Elektronen-Linsen", Optik 2, pp. 114-132, by O. Scherzer, notably pp. 114-119.
The non-rotationally symmetrical optical elements described in the cited article (notably in the sections 1a, 1b and 1c) consist of cylinder lenses and quadrupole fields and monopole fields which act as correction members. The cylinder lenses form an astigmatic path for the electron beams and the correction members, consisting of a combination of quadrupole fields and monopole fields, are arranged in said astigmatic path. The chromatic aberration of the electron path in this known structure is corrected in a first plane which contains the optical axis (the so-called x-z plane), and subsequently the same is done in a second plane which extends perpendicularly thereto (the y-z plane). At the area where the electron path in the x-z plane is subject to chromatic correction, the distance between the electron beam and the optical axis in the y-z plane equals zero for electrons of nominal energy, and vice versa. Because an energy which deviates from the nominal energy occurs in the electron beam, the electron rays having this deviating energy follow a path other than the rays of nominal energy. The electron rays having this deviating energy thus traverse the correction member along a path which deviates from the nominal path; consequently, at said areas with a distance zero from the axis, the distance from the axis is not equal to zero for these rays. However, in the cited article it is assumed that for the rays of deviating energy this distance from the optical axis is so small that the deflecting effect thereof is negligibly small. (In this respect see notably section 1c of the cited article.)
III-1 The Problem Stemming from the State of the Art
Generally speaking, the configuration disclosed in the article by Scherzer cannot be simply used in a particle-optical apparatus. The configuration according to Scherzer constitutes an imaging system, whereas a particle optical apparatus such as a SEM needs a correction system which corrects only the chromatic aberration and has no or hardly any effect on the strength of the imaging lens (the objective). It would be feasible to replace the system known from the cited article by a system with rays which arrive in parallel and emerge in parallel, so that the latter system can be succeeded by the objective and the chromatic aberration of the system can compensate that of the objective. The described assumption that for the rays of deviating energy the distance from the optical axis is so small that the deflecting effect is negligibly small, however, appears to be invalid in the practical circumstances of contemporary particle-optical apparatus, as will be described in detail hereinafter in the sections III-2-a and III-2-b.
Notably in particle-optical apparatus of the SEM type, due to the required strength of the objective lens the electrostatic fields to be used in the correction unit have such a strength that, despite a small distance from the axis, inadmissible influencing of the electron rays traveling at a small distance from the optical axis would still take place. Consequently, electrons having an energy deviating from the nominal value would leave the corrector at a location which deviates substantially from the nominal location. This effect would then introduce a further error, the so-called chromatic magnification error, which would limit the resolution of the particle-optical apparatus to substantially the same extent as the already described chromatic aberration. This is because it can be demonstrated that it holds for the radius of the dispersion circle r.sub.spot due to the latter chromatic error that: r.sub.spot =a(.delta..PHI.).sup.2 C.sub.c.alpha., in which a is a constant, C.sub.c is the coefficient of chromatic aberration of the objective, .delta..PHI. is the relative deviation from the nominal energy of the electrons (so .delta..PHI.=.DELTA..PHI./.PHI..sub.0) and .alpha. is the angle of aperture of the electron beam. According to computer simulations, the constant a is much larger than 100, so that hardly any gain in respect of resolution is made by trading off the first-order chromatic aberration against the latter (second-order) chromatic aberration.
In the following sections III-2-a and III-2-b it will be demonstrated that the use of the method disclosed in the cited article by Scherzer in an electron microscope leads to an unacceptably large chromatic magnification error.
III-2 The Chromatic Magnification Error Incurred While Using the Present State of the Art
First an expression for the relevant optical properties of one correction element will be derived in the following sections III-2-a-1 to III-2-a-3, both for the correction strength (K.sub.corr) in the x-z plane in which the correction element has direct vision (so lens strength K.sub.x =0) and for the lens strength (K.sub.y) in the y-z plane. Comparison of the two quantities K.sub.corr and K.sub.y will reveal the strength in the y-z plane for the desired correction of the (first-order) chromatic aberration. Subsequently, in the next section III-2-b it will be demonstrated that the use of such a correction element in a correction system leads to an unacceptably high value of the chromatic magnification error.
III-2-a Determination of the Relevant Optical Properties of One Correction Element
For the following calculations use is made of the "weak corrector approach", i.e. the variation in the axis potential of the corrector is much smaller than the potential of the accelerating field, so that the perturbation theory can be used; it is also assumed that the chromatic correction takes place in the x-z plane.
The starting point is the equation (1.4) in the cited article by Scherzer, being the paraxial motion equation in the x-z plane:
 ##EQU1##
The various symbols in the expression (1) have the following meaning:
Using the generally known transformation to the reduced co-ordinate X=x.PHI..sup.1/4 (so x=X.PHI..sup.-1/4), in which X is the so-called "Picht variable", a simplified version of the paraxial motion equation (1) is obtained:
X"+TX=0 (2)
The variable T in the expression (2) is a function of .PHI., .PHI.' and .PHI..sub.2 in conformity with: ##EQU2##
This expression (3) for T is obtained by inserting the expression for x as a function of X and its derivatives in the expression (1).
In order to reach a definition of the corrector strength of a correction element for the chromatic aberration, an analogy with a rotationally symmetrical lens is considered. For a rotationally symmetrical lens the coefficient of chromatic aberration C.sub.c is defined in generally known manner in conformity with the equation: ##EQU3##
in which the symbols have the following meaning:
Analogous to the foregoing definition, the corrector strength K.sub.corr of a correction element for the chromatic aberration is defined as: ##EQU4##
or ##EQU5##
The symbols in the expressions (5) and (6) which have not yet been described have the following meaning:
In order to find an expression for the strength of the correction element in the x-z plane, therefore, the deviation of the angle between the electron path and the optical axis .DELTA.X'(1) must be determined. To this end, first the function T (see expression (3)) is explicitly determined, after which, using this explicit expression for T, the quantity .DELTA.X'(1) is determined by solution of the differential equation given by the expression (2).
The perturbation theory is applied so as to execute the above calculations; use can then be made of a series development in which T and X' are not developed further than the second order.
III-2-a-1 Determination of the Function T
The determination of the function T is based on the expression (3). It is assumed that the variation of the potential .PHI. in the correction element is small relative to the energy (expressed in an equivalent potential measure) .PHI..sub.0 with which the electrons enter the correction element. For the overall potential .PHI. of an arbitrary electron in a correction element, the following can then be written: .PHI.=.PHI..sub.0 +.epsilon.g, in which g is a function of z which represents the potential variation in the correction element, and the quantity .epsilon. is an order parameter whose exponent represents the order of the series development. The quantity .PHI..sub.0 is constant, so not a function of z.
The calculation holds for all electrons, so for electrons of nominal potential .PHI..sub.0n as well as for electrons which deviate therefrom by an arbitrary amount .DELTA..PHI., so electrons having the potential .PHI..sub.0n +.DELTA..PHI..
For the function g=g(z) it holds at the entrance (z=-1) and at the exit (z=1) of the correction element that: g(-1)=g'(-1)=g(1)=g'(1)=0. Using said choice for .PHI., it also follows that .PHI.'=.epsilon.g'.
The choice of .PHI..sub.2 is based on the expression (2.6) in the cited article by Scherzer (see page 117). In this expression .PHI.=.PHI..sub.0n +.epsilon.g (the axis potential of an electron of nominal energy), .PHI.'=.epsilon.g' and .PHI."=.epsilon.g" are inserted and developed up to and including the second order. Execution of these operations yields for .PHI..sub.2 : ##EQU6##
This expression (7) is now generalized by substituting a non-specified function h=h(z) for -(1/16)g'.sup.2. The expression (7) thus becomes: ##EQU7##
(This generalization is performed because it would be very difficult in practice to manufacture an apparatus exhibiting the desired potential variation according to the term .PHI.'.sup.2 /.PHI. from the Scherzer formule (2.6). It follows from the calculation, and it has been demonstrated by means of computer simulation, that the real appearance of this function h does not essentially influence the ultimate correction effect, so that this generalization is justified and hence said term need not be realized exactly.)
The expression (8) as well as the previously mentioned expression .PHI.=.PHI..sub.0 +.epsilon.g and its derivatives .PHI.'=.epsilon.g' and .PHI."=.epsilon.g" are then inserted in the expression (3) for the function T, after which this function is developed up to and including the second power of .epsilon.. This yields the following expression for the function T: ##EQU8##
This expression (9) consists of a first-order term T.sub.1 (so proportional to .epsilon.g) and a second-order term T.sub.2 (so proportional to .epsilon..sup.2). Therefore, if T=T.sub.1 +T.sub.2 is written, it holds that: ##EQU9##
III-2-a-2 Determination of the Deviation of the Exit Angle .DELTA.X'(1)
In order to determine the exit angle .DELTA.X'(1), hereinafter first the function h will be determined in such a manner that the correction element has direct vision (as already indicated in section III-2); this will result in the expression (18). Subsequently, the strength of the correction element is determined as a function of the potential at the axis .PHI.(z); this will yield the expression (20).
The path equation X is now written as the sum of three terms, i.e. a zero-order term which is equal to 1, a first-order term X.sub.1 and a second-order term X.sub.2, so X=1+X.sub.1 +X.sub.2. (It also follows therefrom that X'=X.sub.1 '+X.sub.2 ' and X"=X.sub.1 "+X.sub.2 ".) The differential equation for X as given by the expression (2) assumes the following appearance after insertion of the above expressions X=1+X.sub.1 +X.sub.2 and X"=X.sub.1 "+X.sub.2 " and the previously mentioned expression T=T.sub.1 +T.sub.2 : EQU X.sub.1 "+X.sub.2 "+(T.sub.1 +T.sub.2)(1+X.sub.1 +X.sub.2)=0 (12)
which, after splitting into the first-order part and the second-order part and insertion of expression (10) for T.sub.1, yields for X.sub.1 : ##EQU10##
As the second-order part of the expression (12) it follows for X.sub.2 that: EQU X.sub.2 "+T.sub.1 X.sub.1 +T.sub.2 =0 (14)
insertion of the expressions (10) for T.sub.1, (11) for T.sub.2 and (13) for X.sub.1, then yields: ##EQU11##
It is to be noted that it appears from the expression (15) that X.sub.2 " is proportional to .epsilon..sup.2, as could be expected because X.sub.2 represents the second-order part of the path equation. The above differential equation for X.sub.2 can then be integrated once over the interval between z=-1 and z=1, yielding: ##EQU12##
for which use is made of ##EQU13##
which expression (17) follows from partial integration with secondary conditions g(-1)=g'(-1)=g(1)=g'(1)=0. X.sub.2 '(-1) in the expression (17) equals zero, because it holds that X'(-1)=X.sub.1 '(-1)+X.sub.2 '(-1), where X'(-1) equals zero because of the secondary condition g'(-1)=0 in combination with the expression (13), and where X.sub.1 '(-1) equals zero because of the secondary condition g'(-1)=0 in combination with the expression (13).
The form .intg.hdz can also be expressed in -.intg.g'.sup.2 dz. To this end, use is made of the requirement that the correction unit must have direct vision for the nominal energy, so X'(1)=0 for .DELTA..PHI.=0 or .PHI..sub.0 =.PHI..sub.0n. Because it holds that X'(1)=X.sub.1 '(1)+X.sub.2 '(1) and X.sub.1 '(1)=0 on the basis of the expression (13) in combination with the secondary condition g'(1)=0, it thus holds that X.sub.2 '(1)=0 for .PHI..sub.0 =.PHI..sub.0n. This means that the left term of the expression (16) must be equal to zero, so that the relation between .intg.hdz and -.intg.g'.sup.2 dz follows from: ##EQU14##
The above integral expression (18) can be substituted for h in the expression (16), whereas .PHI..sub.0 =.PHI..sub.0n +.DELTA..PHI. can be written for an electron of non-nominal potential, and for X.sub.2 '(1)-X.sub.2 '(-1)=.DELTA.X.sub.2 '(1); .PHI.'=.epsilon.g' is also inserted (see section III-2-a-1). This yields: ##EQU15##
The strength of the correction element is given by the expression (6) in which the quantity .DELTA.X.sub.2 '(1) in the numerator is given by the expression (19). As regards the denominator X(1) of the expression (6) it is to be noted that it is of the second order, because X(1)=1+X.sub.1 (1)+X.sub.2 (1), where X.sub.1 (1)=0 (because g(1)=0 in combination with the expression (13)) and where X.sub.2 (1) is of the second order, so that X(1) is of the second order. Because the numerator of the expression (6) is of the second order (see expression (16) in which .epsilon..sup.2 occurs), X(1) in the denominator of the expression (6) can be made equal to 1. In these circumstances, insertion of the expression (19) in the expression (6) yields the ultimate expression concerning the strength K.sub.corr of the correction element in the x-z plane: ##EQU16##
III-2-a-3 Determination of the Strength of the Correction Element in the y-z Plane
The calculation described in this section will hold for an electron of nominal potential .PHI..sub.0n. On the basis of the equation (1.4) in the cited article by Scherzer, being the paraxial motion equation in the y-z plane, in the y-z plane the known transformation to the reduced co-ordinate Y is performed, analogous to the x-z plane: Y=y.PHI..sup.1/4, yielding the paraxial motion equation in the y-z plane Y"+TY=0. (Compare expression (2) for the x-z plane.) The function T now applicable is then: ##EQU17##
It is to be noted that the expression for the function T for the y-z plane is the same, except for the sign of .PHI..sub.2 /.PHI., as that for the function T for the x-z plane.
The strength K.sub.y of the correction element in the y-z plane is defined as ##EQU18##
Y=1+Y.sub.1 +Y.sub.2 can be defined analogously to the calculation for the x-z plane. The first-order term Y.sub.1 at the area 1 (so Y.sub.1 (1)) is zero, so that Y(1)=1+Y.sub.2 (1), in which Y.sub.2 (1) is of the second order, so that the denominator Y(1) of the expression (26) is of the second order. The numerator Y'(1) again is of the second order for the y-z plane, so that terms of an order higher than zero in the numerator may be ignored and the numerator may be assumed to be 1. Thus, for the strength K.sub.y of the correction element in the y-z plane remains: K.sub.y.apprxeq.-Y'(1), so that in conformity with the above relation Y(1)=1+Y.sub.2 (1) it holds that: K.sub.y =-Y.sub.2 '(1).
The expression yet to be determined for Y.sub.2 '(1) has the same structure as the expression for X.sub.2 '(1) (which follows from the equation (16); however, because of the opposite sign in the function T, a number of coefficients in the derivation of the expression for Y.sub.2 '(1) have received a different value. After a calculation which is performed in the same way as that for the x-z plane, ultimately the following expression is obtained for Y.sub.2 ": ##EQU19##
which expression is analogous to the expression (15) which holds for the x-z plane. Using a calculation which is analogous to that used for the x-z plane, finally the following expression follows for Y.sub.2 '(1) from the expression (23): ##EQU20##
which expression is analogous to an expression for the x-z plane which occurs when the expression (15) is integrated once and an integral expression (18) for h is included therein.
Finally, when .epsilon.g'=.PHI.' is inserted in the expression (24), the ultimate expression for the strength K.sub.y of the correction element in the x-z plane is obtained: ##EQU21##
which, while using the expression (20), yields for the value of K.sub.y : EQU K.sub.y =14K.sub.corr (26)
The important conclusion can be drawn from the expression (26) that in the case of one direct vision correction element, the lens strength in the y-z plane is many times greater than the corrector strength in the x-z plane; the value of this factor, therefore, is of the order of magnitude of 14. Using computer simulation with a typical configuration and also using the exact electron paths, it appears that this factor is approximately 20, so that the previously described approaches in any case yield the correct order of magnitude for this factor.
It is to be noted that the quantity K.sub.y indicates a lens strength, so a quantity whose reciprocal value represents a focal distance, and that the quantity K.sub.corr indicates a corrector strength, so a quantity which indicates to what extent an electron of deviating energy exhibits an angular deviation relative to the optical axis upon leaving the correction element. Therefore, the latter quantity is not a lens strength.
III-2-b Use of a Correction Element in a Correction System
It will be demonstrated hereinafter that the chromatic magnification error assumes an inadmissibly high value because K.sub.y.apprxeq.14K.sub.corr. To this end, it is assumed that the single correction element described above forms part of a known system for the correction of chromatic aberration as described in said article by Archard. The correction system described therein consists of a combination of two quadrupole lenses, each having a strength K.sub.Q, and some correction sub-systems, each sub-system consisting of a single correction element as described above which is arranged between two quadrupole lenses having a strength K.sub.Q /2 each. The overall correction system thus consists of a succession of a first quadrupole lens, a first and a second sub-system, and a second quadrupole lens. For the calculation it is assumed that all quadrupole lenses and the single correction element in the correction system are formed as thin elements, that each quadrupole lens has a chromatic aberration coefficient C.sub.c,Q amounting to the reciprocal strength (i.e. 1/K.sub.Q or 2/K.sub.Q), that for the distance d between the first quadrupole lens and the first correction sub-system it holds that d=1/K.sub.Q, which also holds for the distance d between the second correction sub-system, and that for the distance 2d between the two correction sub-systems it holds that 2d=2/K.sub.Q. Only the chromatic magnification error in the x-z plane at the exit of the overall correction system will be calculated, because the chromatic magnification error in the y-z plane is much smaller.
In the case of a thin quadrupole of strength K.sub.Q, the relationship between x.sub.o and x.sub.o ' (the distance between the outgoing ray in the x-z plane and the optical axis or the slope of this ray) on the one side and x.sub.i and x.sub.i ' (the distance between the incident ray in the x-z plane and the axis or the slope of this ray) on the other hand it holds that: ##EQU22##
in which .delta..PHI. is the relative deviation with respect to the nominal value .PHI..sub.0 of the acceleration voltage, so .delta..PHI.=.DELTA..PHI./.PHI..sub.0.
Analogously, the following holds for the relationship between the relevant quantities in the y-z plane: ##EQU23##
Finally, for a displacement of the ray over a distance z without diffraction of the ray it holds analogously that: ##EQU24##
while, evidently, a corresponding relation holds for the corresponding quantities in the y-z plane.
Furthermore, in the x-z plane it holds for the single correction element as described above that: ##EQU25##
whereas for the relationship between the relevant quantities in the y-z plane it holds that: ##EQU26##
In the expressions (30) and (31) the quantity .delta..PHI. has been omitted in the matrix elements which do not contribute to the ultimate expression for x.sub.o and in which .delta..PHI. would occur. This is justified because the present calculation only aims to give an impression of the order of magnitude of the chromatic magnification error and said matrix elements would make only a (negligibly small) contribution of higher order in .delta..PHI. to the ultimate value of x.sub.o. Such ignoring will also take place hereinafter for the elaboration of the matrices.
Using the above expressions (27) and (30), it now follows for a correction sub-system (i.e. a system consisting of a single correction element which is arranged between two quadrupole lenses, each having a strength K.sub.Q /2) in the x-z plane that: ##EQU27##
and, using the expressions (28) and (31), for the y-z plane that: ##EQU28##
For the derivation of the expressions (32) and (33) it has been assumed that the chromatic aberration of all quadrupoles is zero; this assumption is valid if it holds that K.sub.corr &gt;&gt;K.sub.Q. The beam path in the x-z plane for the overall correction system is then found by combining the beam path in the first quadrupole, the first and the second sub-system, and the second quadrupole. It is again assumed that the chromatic aberration of all quadrupoles is zero. This yields: ##EQU29##
Elaboration of the above expression (34) for x.sub.o yields: ##EQU30##
If it holds that K.sub.corr &gt;&gt;K.sub.Q, the expression (35) can be approximated (for K.sub.y.apprxeq.14K.sub.corr, as has already been demonstrated in the preceding section III-2-a-3) by: ##EQU31##
It is to be noted that the expression (36) represents a suitable approximation if K.sub.corr &gt;&gt;K.sub.Q. If K.sub.corr amounts to only a few times (for example, three or more) K.sub.Q, this expression (36) remains an approximation which can be used to demonstrate that the chromatic magnification error becomes inadmissibly large.
From the expression (34) it also follows for x.sub.o ' that: EQU x.sub.o '=4..delta..PHI..K.sub.corr.x.sub.o (37)
It is to be noted that if the above approximation (i.e. that the chromatic aberration of all quadrupoles is zero) is not permissible, the expression (37) will become: EQU x.sub.o '=4.delta..PHI.(K.sub.corr -K.sub.Q) (38)
For suitable operation as a chromatic aberration corrector it is necessary that the ratio of the exit angle x.sub.o ' and the exit height x.sub.o (so x.sub.o '/x.sub.o) is directly proportional to .delta..PHI.; this requirement is satisfied only if the term .delta..PHI.(4K.sub.corr K.sub.y /K.sub.Q.sup.2) in the expression (36) is sufficiently small in relation to 1. In the preceding section III-2-a-3 it has already been demonstrated that K.sub.y.apprxeq.14K.sub.corr, so that this term becomes 56.delta..PHI.(K.sub.corr /K.sub.Q).sup.2. Practical values for the quantities of the latter expression are, for example: .PHI..sub.0 =500V and .DELTA..PHI.=0.25V, so that .delta..PHI.=10.sup.-4 and K.sub.corr =pK.sub.Q, in which p is a proportionality constant. It follows from the expression (38) that p must in any case be larger than 1. For practical reasons usually a value of the order of magnitude of from 3 to 5 will have to be chosen for p. Using these values, it follows that said term is approximately equal to 0.45, so that it is not negligibly small relative to 1. The chromatic magnification error of the overall corrector, therefore, in this known configuration will have such a high value that the correction system proposed by Archard, using the correction units proposed by Scherzer, cannot be used in practice. This is because the expression mentioned for r.sub.spot in section III-2 will then assume such a high value that the gain as regards resolution of the particle lens to be corrected is substantially canceled by the loss in resolution due to the chromatic magnification error.